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Completed polynomial_over_field_dim_one
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@ -7,7 +7,7 @@ import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
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namespace Ideal
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private lemma singleton_chainHeight_one {α : Type} [Preorder α] [Bot α] : Set.chainHeight {(⊥ : α)} ≤ 1 := by
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private lemma singleton_bot_chainHeight_one {α : Type} [Preorder α] [Bot α] : Set.chainHeight {(⊥ : α)} ≤ 1 := by
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unfold Set.chainHeight
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simp only [iSup_le_iff, Nat.cast_le_one]
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intro L h
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@ -67,7 +67,7 @@ lemma polynomial_over_field_dim_one {K : Type} [Nontrivial K] [Field K] : krullD
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unfold height
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rw [sngletn]
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simp only [WithBot.coe_le_one, ge_iff_le]
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exact singleton_chainHeight_one
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exact singleton_bot_chainHeight_one
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· suffices : ∃I : PrimeSpectrum (Polynomial K), 1 ≤ (height I : WithBot ℕ∞)
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· obtain ⟨I, h⟩ := this
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have : (height I : WithBot ℕ∞) ≤ ⨆ (I : PrimeSpectrum (Polynomial K)), ↑(height I) := by
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